Memory Compression of the Galerkin Volume Integral Equations and Coil Modeling for the Electrical Property Mapping of Biological Tissue

Ilias I. Giannakopoulos

Memory Compression of the Galerkin Volume Integral Equations and Coil Modeling for the Electrical Property Mapping of Biological Tissue

Ilias I. Giannakopoulos
Doctoral Thesis
Skolkovo Institute of Science and Technology, 2020

Abstract

The scope of this doctoral dissertation is to study the interactions of electromagnetic (EM) waves and biological tissue in the presence of a strong magnetic field, and demonstrate the utility of novel methods via their application to simulations for the Magnetic Resonance Imaging (MRI) of realistic human head models. In problems related to MRI, the maximum use of the single operating frequency allows the careful design and optimization of fast and robust algorithms of computational electromagnetics (CEM), based on integral equations (IE). Specifically, surface integral equations (SIE) are employed to analyze the radio-frequency (RF) transmit-receive coils of the magnetic resonance (MR) scanner, while volume integral equations (VIE) model the EM interactions of human tissues with finite electrical properties (EP). The fast and accurate estimation of the interactions above is of paramount importance since a poor design of an RF coil might lead to detrimental effects in the quality of the MR image and the safety of the patient, especially in modern ultra-high-field (UHF) scanners.

Specifically, in the first part of this thesis, we present a method of memory footprint reduction for FFT-based, EM VIE formulations. The arising Green's function tensors have low multilinear rank properties, which allows the employment of tensor decompositions (Tucker, Canonical decompositions, and Tensor Train) for their compression, thereby significantly reducing the required memory storage for numerical simulations. Consequently, the compressed components can fit inside a graphical processing unit (GPU) on which highly parallelized computations can vastly accelerate the iterative solution of the arising linear system. Besides, we provide a variety of novel and efficient matrix-vector product methods that maintain the linear complexity of the traditional element-wise product of FFT-based VIE and can provide up to an order of magnitude of acceleration. For the second part, we turn our interest to the non-invasive cross-sectional mapping of the electrical property (EP) distributions of realistic human head models using MR measurements, and the recently introduced Global Maxwell Tomography (GMT). Previous work evaluated GMT using ideal radiofrequency (RF) excitations, while this dissertation aims to assess GMT's performance in simulation, using a realistic RF coil. The designed coil is a transmit-receive array with eight decoupled channels for 7 Tesla head imaging. We calculated the RF transmit field inside inhomogeneous head models for different RF shimming approaches, and used them as input for GMT to reconstruct brain EP. The coil tuning/decoupling remained relatively stable when the coil was load with different head models. The mean error in EP estimation changed from 7.5% to 9.5% and from 4.84% to 7.2% for the relative permittivity and conductivity, respectively, when changing head models without re-tuning the coil. When an SVD-based RF shimming algorithm is applied, in place of excitation with one coil channel at a time, we observed that the reconstruction slightly improves. Despite errors in EP, the prediction of the RF transmit field, and the voxel-wise absorbed power have less than 0.5% mean error over the entire head. Also, GMT could accurately detect a numerically inserted tumor. The results summarized above show that GMT can reliably reconstruct EP in realistic simulated scenarios using a tailored 8-channel RF coil design at 7 Tesla, thus, enabling future in-vivo GMT experiments. The significance of this work is that GMT could provide accurate estimations of tissue EP, which could be used as biomarkers and could enable patient-specific estimation of RF power deposition, which is an unsolved problem for UHF MRI. Finally, there is a costly trade-off between accuracy and time footprint for GMT, and regrettably, the reconstruction requires days to converge, especially for fine resolutions. Thus we investigate deep learning architectures that can vastly accelerate the reconstruction and overcome such impasses. The proposed approach is to train a tensor-to-tensor convolutional neural network that maps the MR measurements to the corresponding EP of tissue-mimicking phantoms.

The novel RF coil designs can aid GMT to provide accurate estimations of tissue EP, which could be used as biomarkers and could enable patient-specific estimation of RF power deposition, which is an unsolved problem for UHF MRI. Moreover, the novel accelerated through GPU programming, tensor decomposition-based methods, will offer more precise and faster biomedical analysis. As a result, both contributions of this thesis could help to exploit the full potential of UHF MRI.